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Deciding Boolean Algebra with Presburger Arithmetic

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Abstract

We describe an algorithm for deciding the first-order multisorted theory BAPA, which combines Boolean algebras of sets of uninterpreted elements (BA) and Presburger arithmetic operations (PA). BAPA can express the relationship between integer variables and cardinalities of unbounded finite sets, and it supports arbitrary quantification over sets and integers. Our motivation for BAPA is deciding verification conditions that arise in the static analysis of data structure consistency properties. Data structures often use an integer variable to keep track of the number of elements they store; an invariant of such a data structure is that the value of the integer variable is equal to the number of elements stored in the data structure. When the data structure content is represented by a set, the resulting constraints can be captured in BAPA. BAPA formulas with quantifier alternations arise when verifying programs with annotations containing quantifiers or when proving simulation relation conditions for refinement and equivalence of program fragments. Furthermore, BAPA constraints can be used for proving the termination of programs that manipulate data structures, as well as in constraint database query evaluation and loop invariant inference. We give a formal description of an algorithm for deciding BAPA. We analyze our algorithm and show that it has optimal alternating time complexity and that the complexity of BAPA matches the complexity of PA. Because it works by a reduction to PA, our algorithm yields the decidability of a combination of sets of uninterpreted elements with any decidable extension of PA. When restricted to BA formulas, the algorithm can be used to decide BA in optimal alternating time. Furthermore, the algorithm can eliminate individual quantifiers from a formula with free variables and therefore perform projection onto a desirable set of variables. We have implemented our algorithm and used it to discharge verification conditions in the Jahob system for data structure consistency checking of Java programs; our experience suggest that a straightforward implementation of the algorithm is effective on nontrivial formulas as long as the number of set variables is small. We also report on a new algorithm for solving the quantifier-free fragment of BAPA.

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References

  1. Ackermann, W.: Solvable Cases of the Decision Problem. North Holland, Amsterdam (1954)

    MATH  Google Scholar 

  2. Arkoudas, K., Zee, K., Kuncak, V., Rinard, M.: Verifying a file system implementation. In: Sixth International Conference on Formal Engineering Methods (ICFEM ’04), vol. 3308 of LNCS. Seattle (2004)

  3. Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P. (eds.) The Description Logic Handbook: Theory, Implementation and Applications. Cambridge University Press, Boston, Massachusetts (2003)

    MATH  Google Scholar 

  4. Barrett, C., Berezin, S.: CVC lite: A new implementation of the cooperating validity checker. In: Alur, R., Peled, D.A. (eds.) Proceedings of the 16th International Conference on Computer Aided Verification (CAV ’04). Lecture Notes in Computer Science, vol. 3114, pp. 515–518. Boston, Massachusetts (2004)

  5. Berman, L.: The complexity of logical theories. Theor. Comp. Sci. 11(1), 71–77 (1980)

    Article  Google Scholar 

  6. Boigelot, B., Jodogne, S., Wolper, P.: An effective decision procedure for linear arithmetic over the integers and reals. ACM Trans. Comput. Logic 6(3), 614–633 (2005)

    Article  MathSciNet  Google Scholar 

  7. Börger, E., Grädel, E., Gurevich, Y.: The Classical Decision Problem. Springer, Berlin Heidelberg New York (1997)

    MATH  Google Scholar 

  8. Bozga, M., Iosif, R.: On decidability within the arithmetic of addition and divisibility. In: FOSSACS’05. Lecture Notes in Computer Science. Springer, Berlin Heidelberg New York (2005)

    Google Scholar 

  9. Bruyére, V., Hansel, G., Michaux, C., Villemaire, R.: Logic and p-recognizable sets of integers. Bull. Belg. Math. Soc. Simon Stevin 1, 191–238 (1994)

    MathSciNet  Google Scholar 

  10. Bultan, T., Gerber, R., Pugh, W.: Model-checking concurrent systems with unbounded integer variables: Symbolic representations, approximations, and experimental results. ACM Trans. Program. Lang. Syst. 21(4), 747–789 (1999)

    Article  Google Scholar 

  11. Cadoli, M., Schaerf, M., Giovanardi, A., Giovanardi, M.: An algorithm to evaluate quantified boolean formulae and its experimental evaluation. J. Autom. Reason. 28(2), 101–142 (2002)

    Article  MathSciNet  Google Scholar 

  12. Cantone, D., Omodeo, E., Policriti, A.: Set Theory for Computing. Springer, Berlin Heidelberg New York (2001)

    MATH  Google Scholar 

  13. Chaieb, A., Nipkow, T.: Generic proof synthesis for Presburger arithmetic. Technical report, Technische Universität München (2003)

  14. Chandra, A.K., Kozen, D.C., Stockmeyer, L.J.: Alternation. J. Assoc. Comput. Mach. 28(1), 114–133 (1981)

    MathSciNet  Google Scholar 

  15. Chin, W.-N., Khoo, S.-C., Xu, D.N.: Extending sized types with collection analysis. In: ACM PEPM’03, San Diego, California (2003)

  16. Cooper, D.C.: Theorem proving in arithmetic without multiplication. In: Meltzer, B., Michie, D. (eds.) Machine Intelligence, vol. 7, pp. 91–100. Edinburgh University Press, London, UK (1972)

    Google Scholar 

  17. Dewar, R.K.: Programming by refinement, as exemplified by the SETL representation sublanguage. ACM Trans. Program. Lang. Syst. 1(1), 27–49 (1979)

    Article  Google Scholar 

  18. Feferman, S., Vaught, R.L.: The first order properties of products of algebraic systems. Fundam. Math. 47, 57–103 (1959)

    MathSciNet  Google Scholar 

  19. Ferrante, J., Rackoff, C.W.: The Computational Complexity of Logical Theories. Lecture Notes in Mathematics, vol. 718 Springer, Berlin Heidelberg New York (1979)

    Google Scholar 

  20. Gordon, M.J.C., Melham, T.F.: Introduction to HOL, a Theorem Proving Environment for Higher-order Logic. Cambridge University Press, UK (1993)

    MATH  Google Scholar 

  21. Henriksen, J., Jensen, J., Jørgensen, M., Klarlund, N., Paige, B., Rauhe, T., Sandholm, A.: Mona: Monadic second-order logic in practice. In: TACAS ’95. Lecture Notes in Computer Science, vol. 1019. Springer, Berlin Heidelberg New York (1995)

    Google Scholar 

  22. Hoare, C.A.R.: An axiomatic basis for computer programming. Commun. ACM 12(10), 576–580 (1969)

    Article  Google Scholar 

  23. Kapur, D.: Automatically generating loop invariants using quantifier elimination. In: IMACS International Conference on Applications of Computer Algebra, Beaumont, Texas (2004)

  24. Klarlund, N., Møller, A., Schwartzbach, M.I.: MONA implementation secrets. In: Proc. 5th International Conference on Implementation and Application of Automata. Lecture Notes in Computer Science. Springer, Berlin Heidelberg New York (2000)

    Google Scholar 

  25. Kozen, D.: Complexity of boolean algebras. Theor. Comput. Sci. 10, 221–247 (1980)

    Article  MathSciNet  Google Scholar 

  26. Kozen, D.: Theory of Computation. Springer, Berlin Heidelberg New York (2006)

    MATH  Google Scholar 

  27. Kuncak, V., Nguyen, H.H., Rinard, M.: An algorithm for deciding BAPA: Boolean algebra with Presburger arithmetic. In: 20th International Conference on Automated Deduction, CADE-20, Tallinn, Estonia (2005)

  28. Kuncak, V., Rinard, M.: On the theory of structural subtyping. Technical Report 879. Laboratory for Computer Science, Massachusetts Institute of Technology (2003a)

  29. Kuncak, V., Rinard, M.: Structural subtyping of non-recursive types is decidable. In: Eighteenth Annual IEEE Symposium on Logic in Computer Science, Ottawa, Canada (2003b)

  30. Kuncak, V., Rinard, M.: The first-order theory of sets with cardinality constraints is decidable. Technical Report 958, MIT CSAIL (2004)

  31. Kuncak, V., Rinard, M.: Decision procedures for set-valued fields. In: 1st International Workshop on Abstract Interpretation of Object-Oriented Languages (AIOOL), Paris, France (2005)

  32. Kuncak, V., Rinard, M.: An overview of the Jahob analysis system: Project goals and current status. In: NSF Next Generation Software Workshop, Rhodes, Greece (2006)

  33. Lahiri, S.K., Seshia, S.A.: The UCLID decision procedure. In: CAV’04. Lecture Notes in Computer Science, vol. 3114. Springer, Berlin Heidelberg New York (2004)

    Google Scholar 

  34. Lam, P., Kuncak, V., Rinard, M.: Generalized typestate checking using set interfaces and pluggable analyses. SIGPLAN Not. 39, 46–55 (2004)

    Article  Google Scholar 

  35. Lam, P., Kuncak, V., Rinard, M.: Generalized typestate checking for data structure consistency. In: 6th International Conference on Verification, Model Checking and Abstract Interpretation, Paris (2005)

  36. Loewenheim, L.: Über Mögligkeiten im Relativkalkül. Math. Ann. 76, 228–251 (1915)

    Google Scholar 

  37. Marnette, B., Kuncak, V., Rinard, M.: On algorithms and complexity for sets with cardinality constraints. Technical report, MIT CSAIL (2005)

  38. Marriott, K., Odersky, M.: Negative boolean constraints. Technical Report 94/203, Monash University (1994)

  39. Møller, A., Schwartzbach, M.I.: The pointer assertion logic engine. In: Conference on Programming Language Design and Implementation, Snowbird, Utah (2001)

  40. Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM Trans. Program. Lang. Syst. 1(2), 245–257 (1979)

    Article  Google Scholar 

  41. Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL: A Proof Assistant for Higher-Order Logic, vol. 2283 of LNCS. Springer, Berlin Heidelberg New York (2002)

    Google Scholar 

  42. Ohlbach, H.J., Koehler, J.: How to extend a formal system with a Boolean algebra component. In: Schmidt, W.B.P.H. (ed.) Automated Deduction. A Basis for Applications, vol. 3, pp. 57–75. Kluwer, Boston, Massachusetts (1998)

    Google Scholar 

  43. Owre, S., Rushby, J.M., Shankar, N.: PVS: A prototype verification system. In: Kapur, D. (ed.) 11th CADE, vol. 607 of LNAI, pp. 748–752 (1992)

  44. Papadimitriou, C.H.: On the complexity of integer programming. J. Assoc. Comput. Mach. 28(4), 765–768 (1981)

    MathSciNet  Google Scholar 

  45. Podelski, A., Rybalchenko, A.: Transition predicate abstraction and fair termination. In: ACM POPL, pp. 132–144. ACM, New York (2005)

    Chapter  Google Scholar 

  46. Presburger, M.: Über die Vollständigkeit eines gewissen Systems der Aritmethik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. In: Comptes Rendus du premier Congrès des Mathématiciens des Pays slaves, Warsawa, pp. 92–101 (1929)

  47. Pugh, W.: The Omega test: A fast and practical integer programming algorithm for dependence analysis. In: Supercomputing ’91: Proceedings of the 1991 ACM/IEEE Conference on Supercomputing, Albuquerque, New Mexico, pp. 4–13 (1991)

  48. Reddy, C.R., Loveland, D.W.: Presburger arithmetic with bounded quantifier alternation. In: ACM STOC, pp. 320–325. ACM, New York (1978)

    Google Scholar 

  49. Revesz, P.: Quantifier-elimination for the first-order theory of boolean algebras with linear cardinality constraints. In: Proc. Advances in Databases and Information Systems (ADBIS’04). Lecture Notes in Computer Science, vol. 3255. Springer, Berlin Heidelberg New York (2004)

    Google Scholar 

  50. Ruess, H., Shankar, N.: Deconstructing Shostak. In: Proc. 16th IEEE LICS, Washington–Brussels–New York (2001)

  51. Rugina, R.: Quantitative shape analysis. In: Static Analysis Symposium (SAS’04), Verona, Italy (2004)

  52. Sagiv, M., Reps, T., Wilhelm, R.: Parametric shape analysis via 3-valued logic. ACM Trans. Program. Lang. Syst. 24(3), 217–298 (2002)

    Article  Google Scholar 

  53. Skolem, T.: Untersuchungen über die Axiome des Klassenkalküls and über “Produktations-und Summationsprobleme”, welche gewisse Klassen von Aussagen betreffen. Skrifter utgit av Vidnskapsselskapet i Kristiania, I, klasse, no. 3, Oslo (1919)

  54. Thatcher, J.W., Wright, J.B.: Generalized finite automata theory with an application to a decision problem of second-order logic. Math. Syst. Theory 2(1), 57–81 (1968)

    Article  MathSciNet  Google Scholar 

  55. Thomas, W.: Languages, automata, and logic. In: Handbook of Formal Languages Vol. 3: Beyond Words. Springer, Berlin Heidelberg New York (1997)

    Google Scholar 

  56. Tinelli, C., Zarba, C.: Combining non-stably infinite theories. J. Autom. Reason. 34(3), 209–238 (2005)

    Article  Google Scholar 

  57. Tiwari, A.: Decision procedures in automated deduction. Ph.D. thesis, Department of Computer Science, State University of New York at Stony Brook (2000)

  58. Voronkov, A.: The anatomy of Vampire (implementing bottom-up procedures with code trees). J. Autom. Reason. 15(2), 237–265 (1995)

    Article  MathSciNet  Google Scholar 

  59. Weidenbach, C.: Combining superposition, sorts and splitting. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. 2, Chapt. 27, pp. 1965–2013. Elsevier Science, Amsterdam (2001)

    Google Scholar 

  60. Yorsh, G., Reps, T., Sagiv, M.: Symbolically computing most-precise abstract operations for shape analysis. In: Proceedings of 10th TACAS. Springer, Berlin Heidelberg New York (2004)

    Google Scholar 

  61. Zarba, C.G.: The combination problem in automated reasoning. Ph.D. thesis, Stanford University (2004a)

  62. Zarba, C.G.: Combining sets with elements. In: Dershowitz, N. (ed.) Verification: Theory and Practice. Lecture Notes in Computer Science, vol 2772, pp. 762–782. Springer, Berlin Heidelberg New York (2004b)

    Google Scholar 

  63. Zarba, C.G.: A quantifier elimination algorithm for a fragment of set theory involving the cardinality operator. In: 18th International Workshop on Unification (2004c)

  64. Zarba, C.G.: Combining sets with cardinals. J. Autom. Reason. 34(1), 1–29 (2005)

    Article  MathSciNet  Google Scholar 

  65. Zhang, L., Malik, S.: Conflict driven learning in a quantified Boolean satisfiability solver. In: ICCAD ’02: Proceedings of the IEEE/ACM International Conference on Computer-Aided Design, pp. 442–449. New York, USA (2002)

    Google Scholar 

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Authors and Affiliations

  1. MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar Street, Cambridge, MA, USA

    Viktor Kuncak & Martin Rinard

  2. Singapore-MIT Alliance, 3 Science Drive 2, Singapore, 117543, Singapore

    Huu Hai Nguyen & Martin Rinard

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  1. Viktor Kuncak
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Kuncak, V., Nguyen, H.H. & Rinard, M. Deciding Boolean Algebra with Presburger Arithmetic. J Autom Reasoning 36, 213–239 (2006). https://doi.org/10.1007/s10817-006-9042-1

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